Journal of Applied Mathematics 
Volume 2 (2002), Issue 3, Pages 131-139
doi:10.1155/S1110757X02111041

Semigroup theory applied to options

D. I. Cruz-Báez and J. M. González-Rodríguez

 Department of Applied Economics, University of La Laguna, Campus de Guajara, s/n, Edificio de Económicas - Empresariales, (Tenerife), La Laguna 38071, Spain
Received 5 November 2001; Revised 5 March 2002

Abstract

Black and Scholes (1973) proved that under certain assumptions about the market place, the value of a European option, as a function of the current value of the underlying asset and time, verifies a Cauchy problem. We give new conditions for the existence and uniqueness of the value of a European option by using semigroup theory. For this, we choose a suitable space that verifies some conditions, what allows us that the operator that appears in the Cauchy problem is the infinitesimal generator of a C0-semigroup T(t). Then we are able to guarantee the existence and uniqueness of the value of a European option and we also achieve an explicit expression of that value.